2 Answers
2

From Proposition 6.8 on pdf page 8 of DUSART, you may take
$$ x_0 = \max \left( 396738, \; e^{\left( \frac{1}{5 \sqrt \delta} \right)} \right). $$
This is not the optimal value of your $x_0 = x_0(\delta)$ but it works.

I don't think that the answer above is correct. In fact, such an $ x_{0} $ does exist. It is a consequence of the Prime Number Theorem. One may refer to the Wikipedia article on Bertrand's Postulate: http://en.wikipedia.org/wiki/Bertrand%27s_Postulate. As for the original question, I'm not sure if bounds have been established for the size of $ x_{0} $, except in certain cases. Examples are also given in the Wikipedia article mentioned above. One of the references listed there, an article by Lowell Schoenfeld, gives the following result: For any $ n \geq 2010760 $, there exists a prime between $ n $ and $ \left( 1 + \frac{1}{16597} \right) n $. I believe that $ 2010760 $ is the smallest $ n_{0} \in \mathbb{N} $ corresponding to $ \delta = \frac{1}{16597} $. However, I haven't read the article yet, so please go ahead and read it to help me verify what I've written here.

I had misread the question; I read $\delta x$ as an additive constant. You're right that it's probably meant to be $\delta$ times $x$. I've deleted my answer. Note that "the answer above" isn't meaningful on this site, since answers are ordered according to criteria that can a) be selected differently by different voters (active/oldest/votes) and b) change over time.
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jorikiSep 30 '12 at 19:41

@daniel, is it possible to find $x_{0}$ for any $\delta$?
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Lindsay DuranSep 30 '12 at 23:17

@LindsayDuran: Sorry, the result in the answer about Schoenfeld is not conditional. There are conditional results in the paper but I think this result is from studying the zeros of the zeta function very carefully. It's really a three-paper series.
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danielOct 1 '12 at 1:14